# Absolute Value equations in 2 variables (both x and y ) (a relation based on absolute)

I came across equations in a math book that contains absolute of both x and y. I have done many complicated absolute equations and inequalities, but with only the x being in absolute bars. I don't even know how to start off to analyze a situation of this type.

Here are some examples:

1) |x| + |y+2| = 1
2) |x| - |y| = 1
3) x = 4 - |y|

Hope someone can help with these type of equations.

## 1 Answer

Try analyzing the equations by using symmetries. For example, in $|x| - |y| = 1$, you may try to take away the absolute values first if you are uncomfortable with them, or place it only in one variable (your call, really).

So we have $|x| - y = 1$, which is the same as $y = |x| - 1$. What is the graph of this? If I replace $x$ with $-x$, it really doesn't change the equation, does it? So what does this mean (in terms of reflecting the graph on some axis?)

Now you can try for $y$ and $-y$, use symmetries and you can generate a graph of the equation $|x| - |y| = 1$.

Finally, for $|x| - |y-4| = 1$ for example, just take note this is a translation 4 units right. So there is really no need to graph it all over again!

• OK, I see. So we can first reduce to y as a function of |x|, since we know how to graph the one variable case. But we generate this +y, for y>0 case and -y for y<0, just using the fundamental definition of absolute. OK, I get it now. Thanks to everyone for the great help. Really appreciate it!!! – Palu Oct 8 '15 at 5:58